Problem: On her way home from the laboratory, Duru realized that she left a test tube containing $50{,}000$ bacteria in the lab. Each minute that passes, $\dfrac{1}{3}$ of the total number of bacteria duplicate. If the number of bacteria reaches $100{,}000$, the test tube will explode! Naturally, she turned around and rushed back to the lab. It took Duru $t$ minutes to return to the lab, and she found the test tube intact. Write an inequality in terms of $t$ that models the situation.
The strategy This problem involves bacteria duplicating. Specifically, every minute, the number of bacteria will increase by a factor of $\dfrac43$. [Why?] So, to find the number of bacteria in the test tube over time, we repeatedly multiply the original number of bacteria, $50{,}000$, by $\dfrac43$. Because of this, we know we can model the situation with an exponential expression of the form $ab^x$, where $a$ is $50{,}000$ and $b$ is $\dfrac43$. We now only need to find $x$, which represents the number of times $\dfrac{1}{3}$ of the total number of bacteria duplicate. Finding the exponent We know that each minute that passes, $\dfrac{1}{3}$ of the bacteria duplicate. Since time is in minutes, the exponent is simply $t$. Writing an inequality We can now replace $x$ in the original model with $t$. Therefore, the expression $50{,}000\cdot \left(\dfrac43\right)^{t}$ models the number of bacteria in the test tube after $t$ minutes, or the number of bacteria in the test tube when Duru returned to the lab. Since the test tube was still intact when Duru returned to the lab, we know that the number of yeast cells in the test tube must be less than $100{,}000$. Therefore, we have the following inequality. $50{,}000\cdot \left(\dfrac43\right)^{t}<100{,}000$ The answer An inequality that models the situation is $50000\cdot \left(\dfrac43\right)^{t}<100000$.